The Gemara learns from the verse that states vhoyisa ach sameach, and you shall be nothing but joyous, that there is an obligation for one to be joyous on the night of Shemini Atzeres. There is a contradiction in the words of Rashi if this obligation also applies to Shemini Atzeres by day. The commentators ask that we usually find that the word ach, literally translated as but, is used to exclude something, whereas in this instance, the word ach is used to include something.
It is said in the name of the Vilna Gaon that on the first days of Sukkos, besides the mitzvah of being joyous, there are additional mitzvos of sitting in the Sukkah and taking the lulav. On Shemini Atzeres, however, there is only the mitzvah of being joyous.Shearim Mitzuyanim B’Halacha adds that this can explain the language used in the Gemara that the verse is including the obligation to be joyous on Shemini Atzeres, although the verse is in essence excluding the other two mitzvos of Sukkah and lulav. The explanation for this is that since the other two mitzvos do not apply on Shemini Atzeres, there is more time to fulfill the mitzvah of being joyous.